Derivatives and Anti-derivatives

Question 1

Derivative of a constant k

Answer 1

d/dx[k] = 0

Question 2

Derivative of x

Answer 2

d/dx[x] = 1

Question 3

Derivative of x^n

Answer 3

d/dx[x^n] = nx^(n-1)

Question 4

Derivative of e^x

Answer 4

d/dx[e^x] = e^x

Question 5

Derivative of a^x

Answer 5

d/dx[a^x] = a^x · ln(a)

Question 6

Derivative of ln(x)

Answer 6

d/dx[ln(x)] = 1/x

Question 7

Derivative of log_a(x)

Answer 7

d/dx[log_a(x)] = 1/(x·ln(a))

Question 8

Derivative of sin(x)

Answer 8

d/dx[sin(x)] = cos(x)

Question 9

Derivative of cos(x)

Answer 9

d/dx[cos(x)] = -sin(x)

Question 10

Derivative of tan(x)

Answer 10

d/dx[tan(x)] = sec^2(x)

Question 11

Derivative of cot(x)

Answer 11

d/dx[cot(x)] = -csc^2(x)

Question 12

Derivative of sec(x)

Answer 12

d/dx[sec(x)] = sec(x)·tan(x)

Question 13

Derivative of csc(x)

Answer 13

d/dx[csc(x)] = -csc(x)·cot(x)

Question 14

Derivative of arcsin(x)

Answer 14

d/dx[arcsin(x)] = 1/√(1-x^2)

Question 15

Derivative of arccos(x)

Answer 15

d/dx[arccos(x)] = -1/√(1-x^2)

Question 16

Derivative of arctan(x)

Answer 16

d/dx[arctan(x)] = 1/(1+x^2)

Question 17

Derivative of arccot(x)

Answer 17

d/dx[arccot(x)] = -1/(1+x^2)

Question 18

Derivative of arcsec(x)

Answer 18

d/dx[arcsec(x)] = 1/(|x|·√(x^2-1))

Question 19

Derivative of arccsc(x)

Answer 19

d/dx[arccsc(x)] = -1/(|x|·√(x^2-1))

Question 20

Derivative of sinh(x)

Answer 20

d/dx[sinh(x)] = cosh(x)

Question 21

Derivative of cosh(x)

Answer 21

d/dx[cosh(x)] = sinh(x)

Question 22

Derivative of tanh(x)

Answer 22

d/dx[tanh(x)] = sech^2(x)

Question 23

Sum rule

Answer 23

d/dx[f(x) + g(x)] = f’(x) + g’(x)

Question 24

Product rule

Answer 24

d/dx[f(x)·g(x)] = f’(x)·g(x) + f(x)·g’(x)

Question 25

Quotient rule

Answer 25

d/dx[f(x)/g(x)] = [f’(x)·g(x) - f(x)·g’(x)]/[g(x)]^2

Question 26

Chain rule

Answer 26

d/dx[f(g(x))] = f’(g(x))·g’(x)

Question 27

Anti-derivative of x^n (n≠-1)

Answer 27

∫x^n dx = x^(n+1)/(n+1) + C

Question 28

Anti-derivative of 1/x

Answer 28

∫(1/x) dx = ln|x| + C

Question 29

Anti-derivative of e^x

Answer 29

∫e^x dx = e^x + C

Question 30

Anti-derivative of a^x

Answer 30

∫a^x dx = a^x/ln(a) + C

Question 31

Anti-derivative of sin(x)

Answer 31

∫sin(x) dx = -cos(x) + C

Question 32

Anti-derivative of cos(x)

Answer 32

∫cos(x) dx = sin(x) + C

Question 33

Anti-derivative of tan(x)

Answer 33

∫tan(x) dx = -ln|cos(x)| + C

Question 34

Anti-derivative of sec^2(x)

Answer 34

∫sec^2(x) dx = tan(x) + C

Question 35

Anti-derivative of sec(x)tan(x)

Answer 35

∫sec(x)tan(x) dx = sec(x) + C

Question 36

Anti-derivative of 1/√(1-x^2)

Answer 36

∫1/√(1-x^2) dx = arcsin(x) + C

Question 37

Anti-derivative of 1/(1+x^2)

Answer 37

∫1/(1+x^2) dx = arctan(x) + C

Question 38

Anti-derivative of 1/(x√(x^2-1))

Answer 38

∫1/(x√(x^2-1)) dx = arcsec(x) + C

Question 39

Basic u-substitution concept

Answer 39

∫f(g(x))·g’(x) dx = ∫f(u) du where u = g(x)

Question 40

Integration by parts formula

Answer 40

∫u·dv = u·v - ∫v·du

Question 41

Anti-derivative of 1/(a^2+x^2)

Answer 41

∫1/(a^2+x^2) dx = (1/a)·arctan(x/a) + C

Question 42

Anti-derivative of 1/√(a^2-x^2)

Answer 42

∫1/√(a^2-x^2) dx = arcsin(x/a) + C

Question 43

Anti-derivative of 1/√(x^2+a^2)

Answer 43

∫1/√(x^2+a^2) dx = ln|x + √(x^2+a^2)| + C

Question 44

Anti-derivative of 1/(x^2-a^2)

Answer 44

∫1/(x^2-a^2) dx = (1/2a)·ln|(x-a)/(x+a)| + C

Question 45

Anti-derivative of sinh(x)

Answer 45

∫sinh(x) dx = cosh(x) + C

Question 46

Anti-derivative of cosh(x)

Answer 46

∫cosh(x) dx = sinh(x) + C

Question 47

Anti-derivative of sech^2(x)

Answer 47

∫sech^2(x) dx = tanh(x) + C

Question 48

First Fundamental Theorem of Calculus

Answer 48

d/dx[∫_a^x f(t) dt] = f(x)

Question 49

Second Fundamental Theorem of Calculus

Answer 49

∫_a^b f(x) dx = F(b) - F(a) where F’(x) = f(x)